# A coupled compressible two-phase flow with the biological dynamics modeling the anaerobic biodegradation process of waste in a landfill

2022;
: pp. 483–500

Accepted: March 23, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 483–500 (2022)

Authors:
1
University of Haute-Alsace, University of Strasbourg, France
2
Ibn Tofail University, Equipe d’Ingénierie Mathématique (EIMA), Laboratory: EDP, Algèbre et Géométrie Spectrale, Kénitra, Morocco
3
University of Haute-Alsace, France; Ibn Tofail University, Equipe d’Ingénierie Mathématique (EIMA), Laboratory: EDP, Algèbre et Géométrie Spectrale, Kénitra, Morocco

In this article, we present and study a new coupled model combining the biological and the mechanical aspects describing respectively the process of the biogas production and the compressible two-phase leachate-biogas flow during the anaerobic biodegradation of organic matters in a landfill, which is considered a reactive porous medium.  The model obtained is governed by a reaction-diffusion system for the bacterial activity coupled with a compressible two-phase flow system of a non-homogeneous porous medium.  We carry out the analysis and the numerical approximation of the model within a variational framework.  We propose a full discrete system based on a second-order BDF-time scheme and P1-conforming finite element and we derive an efficient algorithm for the coupled system.  We perform some numerical simulations in 2D and 3D examples in agreement with the theoretical analysis.

1. Belhachmi Z., Mghazli Z., Ouchtout S.  Mathematical modeling and numerical approximation of a leachate flow in the anaerobic biodegradation of waste in a landfill.  Mathematics and Computers in Simulation. 185, 174–193 (2021).
2. Hassam S., Ficara E., Leva A., Harmand J.  A generic and systematic procedure to derive a simplified model from the anaerobic digestion model No. 1 (ADM1).  Biochemical Engineering Journal. 99, 193–203 (2015).
3. Hénon F., Debenest G., Lefevre X., Pommier S., Chenu D., Quintard M.  Simulation of Transport and Impact of Moisture Content on the Biodegradation.  Fourth International Workshop "Hydro-Physico-Mechanics of Landfills". Santander, Spain (2011).
4. Gallo C., Manzini G.  A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation kinetics.  Communications in Numerical Methods in Engineering. 17 (5), 325–336 (2001).
5. Chenu D.  Modélisation des transferts réactifs de masse et de chaleur dans les installations de stockage de déchets ménagers: application aux installations de type bioréacteur.  Doctoral dissertation, Institut National Polytechnique de Toulouse (2007).
6. Ahusborde E., Amaziane B., El Ossmani M.,  Moulay M.  Numerical Modeling and Simulation of Fully Coupled Processes of Reactive Multiphase Flow in Porous Media.  J. Math. Study. 52 (4), 359–377 (2019).
7. Pohland F. G., Al-Yousfi B.  Design and Operation of Landfills for optimum stabilization and biogaz production.  Water Science & Technology. 30 (12), 117–124 (1994).
8. Harmand J., Lobry C., Rapaport A., Sari T.  The Chemostat: Mathematical Theory of Microorganisms Cultures. ISTE Wiley (2017).
9. Smith H. L., Waltman P.  The theory of the chemostat: dynamics of microbial competition (Vol. 13).  Cambridge University Press (1995).
10. Dollé G., Duran O., Feyeux N., Frénod E., Giacomini M., Prud'Homme C.  Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++.  ESAIM: Proceedings and Surveys. 55, 83–110 (2016).
11. Rouez M.  Dégradation anaérobie de déchets solides: Caractérisation, facteurs d'influence et modélisations. Laboratoire de Génie Civil et d'Ingénierie Environnementale.  Lyon, Institut National des Sciences Appliquées Docteur, 259 (2008).
12. Fekih-Salem R., Harmand J., Lobry C., Rapaport A., Sari T.  Extensions of the chemostat model with flocculation.  Journal of Mathematical Analysis and Applications. 397 (1), 292–306 (2013).
13. Rapaport A., Nidelet T., El Aida S., Harmand J.  About biomass overyielding of mixed cultures in batch processes.  Mathematical Biosciences. 322, 108322 (2020).
14. Rapaport A., Nidelet T., Harmand  J.  About biomass overyielding of mixed cultures in batch processes.  8th IFAC Conference on Foundations of Systems Biology in Engineering (FOSBE), Valencia, Spain, 15–18 Oct. (2019).
15. Gnativ Z. Ya., Ivashchuk O. S., Hrynchuk Yu. M., Reutskyi V. V., Koval I. Z., Vashkurak Yu. Z.  Modeling of internal diffusion mass transfer during filtration drying of capillary-porous material.  Mathematical Modeling and Computing. 7 (2), 219–227 (2020).
16. Dimitrova N., Krastanov M.  Model-based optimization of biogas production in an anaerobic biodegradation process.  Computers and Mathematics with Applications. 68 (9), 986–993 (2014).
17. Bernard O., Hadj-Sadok Z., Dochain D., Genovesi A., Steyer J. P.  Dynamical model development and parameter identification for an anaerobic wastewater treatment process.  Biotechnology and Bioengineering. 75 (4), 424–438 (2001).
18. Benyahia B., Sari T., Cherki B., Harmand J.  Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes.  Journal of Process Control. 22 (6), 1008–1019 (2012).
19. Didi I., Dib H., Cherki B.  A Luenberger-type observer for the AM2 model.  Journal of Process Control. 32, 117–126 (2015).
20. Arzate J. A., Kirstein M., Ertem F. C., Kielhorn E., Ramirez Malule H., Neubauer P., Cruz-Bournazou M. N., Junne S.  Anaerobic digestion model (AM2) for the description of biogas processes at dynamic feedstock loading rates.  Chemie Ingenieur Technik. 89, 686–695 (2017).
21. Hmissi M., Harmand J., Alcaraz-Gonzalez V., Shayeb H.  Evaluation of alkalinity spatial distribution in an up-flow fixed bed anaerobic digester.  Water Science and Technology. 77 (4), 948–959 (2018).
22. Abaali M., Harmand J.,  Mghazli Z.  Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media.  Processes. 8 (8), 890 (2020).
23. Pinder G. F., Gray W. G.  Essentials of multiphase flow and transport in porous media. John Wiley and Sons (2008).
24. Agostini F., Sundberg C.,  Navia R.  Is biodegradable waste a porous environment? A review.  Waste Management and Research. 30 (10), 1001–1015 (2012).
25. Ouchtout S., Mghazli Z., Harmand J., Rapaport A., Belhachmi Z.  Analysis of an anaerobic digestion model in landfill with mortality term.  Communications on Pure and Applied Analysis. 19 (4), 2333–2346 (2020).
26. Shi J., Wu Y., Zou X.  Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction-Diffusion Chemostat Model.  Journal of Dynamics and Differential Equations. 32 (2), 1085–1112 (2020).
27. Nguyen-Ngoc D., Leye B., Monga O., Garnier P., Nunan N.  Modeling microbial decomposition in real 3D soil structures using partial differential equations.  International Journal of Geosciences. 4 (10A), 15–26 (2013).
28. Vanrolleghem P. A., Dochain D.  Dynamical Modelling and Estimation in Wastewater Treatment Processes.  IWA Publishing (2001).
29. Hecht F.  New development in FreeFem++.  Journal of numerical mathematics. 20 (3–4), 251–266 (2012).
30. Lanini S.  Analyse et modélisation des transferts de masse et de chaleur au sein des décharges d'ordures ménagères.  Doctoral dissertation, Institut National Polytechnique de Toulouse (1998).
31. Bellenfant G.   Modélisation de la production de lixiviat en centre de stockage de déchets ménagers.  Doctoral dissertation, Institut National Polytechnique de Lorraine-INPL (2001).
32. Aran C.  Modélisation des Ecoulements de Fluides et des Transferts de Chaleur au Sein des Déchets Ménagers.  Application à la Réinjection de Lixiviat dans un Centre de Stockage.  Ph.D. thesis, Institut National Polytechnique de Toulouse, France (2001).
33. Helmig R.  Multiphase flow and transport processes in the subsurface: a contribution to the modeling of hydrosystems.  Springer-Verlag (1997).
34. Chen Z., Huan G., Ma Y.  Computational methods for multiphase flows in porous media (Vol. 2). Siam (2006).
35. Kindlein J., Dinkler D., Ahrens H.  Numerical modeling of multiphase flow and transport processes in landfills.  Waste Management and  Research. 24 (4), 376–387 (2006).
36. Gabbouhy M., Mghazli Z.   Un résultat d'existence de solution faible d'un système parabolique-elliptique non linéaire doublement dégénéré.  Annales de la Faculté des sciences de Toulouse: Mathématiques. 10 (3), 533–546 (2001).
37. Azaïez M., Deville M., Mund E. H.   Eléments finis pour les fluides incompressibles.  PPUR Presses polytechniques (2011).
38. Lichtner P. C.  Continuum formulation of multicomponent-multiphase reactive transport.  Reviews in Mineralogy & Geochemistry. 34, 1–82 (1996).
39. Gholamifard S.  Modélisation des écoulements diphasiques bioactifs dans les installations de stockage de déchets.  Doctoral dissertation, Université Paris-Est (2009).
40. Van Genuchten M. T.  A closed-form equation for predicting the hydraulic conductivity of unsaturated soils.  Soil Science Society of America Journal. 44 (5), 892–898 (1980).
41. Campbell G. S.  A simple method for determining unsaturated conductivity from moisture retention data.  Soil Science. 117 (6), 311–314 (1974).
42. Brooks R. H., Corey A. T.  Hydraulic Properties of Porous Media.  Colorado State Univ. Hydrology Paper No. 3 (1964).
43. Bothe D., Fischer A., Pierre M., Rolland G.  Global wellposedness for a class of reaction-advection-anisotropic-diffusion systems.  Journal of Evolution Equations. 17 (1), 101–130 (2017).